Difference between revisions of "2002 Indonesia MO Problems/Problem 3"
Rockmanex3 (talk | contribs) m (→Solution) |
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Multiply the second equation by the first equation to get | Multiply the second equation by the first equation to get | ||
<cmath>x^3 + xy^2 + xz^2 + x^2y + y^3 + yz^2 + x^2z + y^2z + z^3 = 72</cmath> | <cmath>x^3 + xy^2 + xz^2 + x^2y + y^3 + yz^2 + x^2z + y^2z + z^3 = 72</cmath> | ||
− | Subtract the third equation | + | Subtract the third equation to get |
<cmath>xy^2 + xz^2 + x^2y + yz^2 + x^2z + y^2z = 48</cmath> | <cmath>xy^2 + xz^2 + x^2y + yz^2 + x^2z + y^2z = 48</cmath> | ||
Cube the first equation to get | Cube the first equation to get |
Revision as of 10:33, 15 July 2018
Problem
Find all real solutions from the following system of equations:
Solution
Square the first equation to get
Subtract the second equation from the result to get
Multiply the second equation by the first equation to get
Subtract the third equation to get
Cube the first equation to get
If
,
, and
, the solution triplet is the roots of the polynomial
Factor the polynomial to get
Since
is a triple root to the polynomial, the only solution to the system of equations is
, and plugging the values back in satisfies the system.